Recent developments in the mathematics of diffraction

Moody, Robert V.

doi:10.1524/zkri.2008.1084

Cited by 8 publications

(12 citation statements)

References 19 publications

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“…Writing the ratio as exp ð2pi cðyÞÞ, the phase function c is not defined at these points. Moreover, as one can check explicitly, it does not satisfy the additivity property cðy þ y 0 Þ ¼ cðyÞ þ cðy 0 Þ mod 1, wherefore the ratio fails to be a character on R 2 by violating both defining properties; compare [15] and references therein. Note that an analogous phenomenon already shows up in the comparison of P i with ÀP i , because these windows are not inversion symmetric up to translations.…”

Section: Homometry Of Model Setsmentioning

confidence: 99%

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Homometric point sets and inverse problems

Grimm¹,

Baake

2008

Zeitschrift Für Kristallographie

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Abstract. The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure.Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron's covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane.The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.

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Section: Homometry Of Model Setsmentioning

confidence: 99%

“…However, as discussed in [15], this ambiguity can be resolved with the knowledge of the 3-point correlations. This statement is immediate in our example (via the existence or non-existence of certain patches), but holds in full generality for regular model sets with Euclidean internal space; see [15] and references therein.…”

Section: Homometry Of Model Setsmentioning

confidence: 99%

Homometric point sets and inverse problems

Grimm¹,

Baake

2008

Zeitschrift Für Kristallographie

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“…Separated nets (also called Delone sets) occur as prominent features in the theories of quasiperiodic functions and mathematical quasicrystals. For surveys of these connections the reader is encouraged to read [2,12,13].…”

Section: Introductionmentioning

confidence: 99%

Equivalence classes of codimension-one cut-and-project nets

Haynes

2014

Ergod. Th. Dynam. Sys.

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Abstract. We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement to lattices. Our proof demonstrates that for any irrational α, regardless of Diophantine type, there is a collection of intervals in R/Z which is closed under translation, contains intervals of arbitrarily small length, and along which the discrepancy of the sequence {nα} is bounded above uniformly by a constant.

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“…To provide a rigorous mathematical explanation of the observation that certain non-periodic media (quasicrystals) show sharp Bragg peaks in their X-ray diffraction, mathematicians came up with the notion of a pure point diffractive set. This is a point set Λ of R d to which can be associated an autocorrelation measure γ whose Fourier transformγ is a pure point measure [16] (see also [31] and references therein). In the model of the material by the point set, the Bragg peaks correspond exactly to the points in the dual space R d * that have strictly positivê γ-measure.…”

Section: Introductionmentioning

confidence: 99%

Meyer sets, topological eigenvalues, and Cantor fiber bundles

Kellendonk

Sadun

2013

Journal of the London Mathematical Society

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The diffraction spectrum of a point pattern is very closely related to the dynamical spectrum of an associated dynamical system. This dynamical spectrum is invariant under topological conjugacy and measurable conjugacy, and in particular under a large class of shape deformations. Using measure theory and topology, we construct a pure-point diffractive set, with finite local complexity, that is not a Meyer set. This provides a counterexample to a famous conjecture of Lagarias.

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Recent developments in the mathematics of diffraction

Abstract: Abstract. This is a survey of some recent results in the mathematics of diffraction, concentrating on the use of dynamical systems and stochastic point processes.

Cited by 8 publications

References 19 publications

Homometric point sets and inverse problems

Homometric point sets and inverse problems

Equivalence classes of codimension-one cut-and-project nets

Meyer sets, topological eigenvalues, and Cantor fiber bundles

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